# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

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### Heights for rational points via Neron models

I only just started reading about heights in arithmetic geometry, so forgive me if this question is naive.
Suppose $E$ is an elliptic curve over a number field $K$ with ring of integers $R$ and let $\...

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### Connection Between Knot Theory and Number Theory

Is there any connection between knot theory and number theory in any aspects?
Does anybody know any book that is about knot theory and number theory?

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### Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$

My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular:
Given a projective variety $V$ defined over $L$ algebraically closed, of ...

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### Extending section of étale morphism of adic spaces

This question is related to Lifting points via étale morphism of adic spaces.
Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...

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### On a refinement of Mordell's conjecture for curves

Let $C$ be an algebraic curve of genus $g \geq 2$, defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. It is then defined over a finite extension $K$ of $\mathbb{Q}$. We assume that $C(K) \ne \...

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### Down to earth, intuition behind a Anabelian group [closed]

An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center.
I would like to know ...

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### The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$.
I know that to construct the Jacobian variety associated to $C$, one ...

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### Igusa curve at infinite level

In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....

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### Smooth proper variety over a number field with prescribed bad reductions

Given a number field $K$, and a finite set $S$ containing finite places of $K$. When can we find a smooth proper geomerically connected variety $X$ over $K$ such that $X$ has good reduction outside $S$...

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### What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?

Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ?
For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...

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### On the existence of nice hypercovers

Let $\mathcal{C}$ be a site and $X$ a sheaf of sets on $\mathcal{C}$.
Then there exists a hypercover $K_{\bullet}$ of $X$ such that $K_n$ is a coproduct of representable presheaves on $\mathcal{C}$ ...

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### Nearby cycles and extension by zero

Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$.
Call $i_s ...

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### Smooth proper variety over $\mathbb Q$ with everywhere bad reduction

$\newcommand{\Spec}{\operatorname{Spec}}$
Cross-post from Math.SE, hopefully people more knowledgeable in the field will see the question here on MO.
It is a well-known fact that a smooth projective ...

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### Berthelot’s comparison theorem and functoriality

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.
Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...

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### p-adic Hodge theory for singular projective varieties

In p-adic Hodge theory, one has comparison theorems relating, for example, the crystalline cohomology of the special fiber of a smooth proper family with the etale cohomology of the rigid-analytic ...